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La q-conjecture de Macdonald-Morris pour $G\sb 2$. (The q-conjecture of Macdonald-Morris for $G\sb 2)$. (French) Zbl 0598.05006
The author proves the Macdonald-Morris conjecture for the value of the coefficient of $e\sp 0$ in $$ \prod '(\prod\sp{a}\sb{i=1}(1- e\sp{\alpha}q\sp{i-1})(1-e\sp{-\alpha}q\sp i)),\quad \prod ''(\prod\sp{b}\sb{i=1}(1-e\sp{\beta}q\sp{i-1})(1-e\sp{-\beta}q\sp i)) $$ where $\prod '$ is a product over the positive long roots of $G\sb 2$ and $\prod ''$ is a product over the positive short roots of $G\sb 2$. The proof derives this identity from the evaluation of a ”q-Selberg multidimensional beta integral”, an evaluation found independently by {\it L. Habsieger} [ibid. 302, 615-617 (1986)] and K. Kadell. The derivation of the $G\sb 2$ result from the q-Selberg evaluation was independently realized in the same week by D. Zeilberger.
Reviewer: D.M.Bressoud

05A15Exact enumeration problems, generating functions
17B20Simple, semisimple, reductive Lie (super)algebras
33B15Gamma, beta and polygamma functions